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 {\large {\bf Monogamy property of quantum discord in  photon added \\
  Glauber coherent states of ${\rm GHZ}$-type}}\\
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{\bf  M. Daoud}$^{a,b,c}${\footnote { email: {\sf m$_-$daoud@hotmail.com}}}, {\bf
W. Kaydi}$^{d,e}$ {\footnote { email: {\sf kaydi.smp@gmail.com}}} and {\bf
	 H. El Hadfi }$^{d,e}$ {\footnote { email: {\sf hanane.elhadfi@gmail.com}}}

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$^{a}${\it Max Planck Institute for the Physics of Complex Systems, Dresden, Germany}\\[0.5em]
$^{b}${\it Abdus Salam  International Centre for Theoretical Physics, Miramare, Trieste, Italy}\\[0.5em]
$^{c}${\it Department of Physics , Faculty of Sciences, University Ibnou Zohr,
 Agadir,
Morocco}\\[0.5em]
$^{d}${\it LPHE-Modeling and Simulation, Faculty  of Sciences,
University
Mohammed V, Rabat, Morocco}\\[0.5em]
$^{e}${\it Centre of Physics and Mathematics (CPM),
University
Mohammed V, Rabat, Morocco}\\[0.5em]



\vspace{3cm} {\bf Abstract}
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We investigate the influence of photon excitations on quantum
correlations in tripartite Glauber coherent states of Greenberger-Horne-Zeilinger type.
The pairwise correlations are measured by means of the entropy-based  quantum discord. We analyze
the monogamy property of quantum discord in this class of tripartite states in terms of the strength of Glauber
coherent states and the photon excitation order.


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\section{Introduction}
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\section{Introduction}

Increasingly in the field of quantum information, aspects of entanglement \cite{ref1}, and of other quantum correlations such as, for instance, ``quantum discord" \cite{ref2}, between two qubits have been described for a class of pure and mixed states that have come to be called ``$X$-states" \cite{ref3}. Although their use goes back further \cite{ref4}, they were so named in \cite{ref3} because of the visual appearance of the density matrix, that it looks like the letter in the alphabet:

\begin{equation}
\rho =  \left(
\begin{array}{cccc}
\rho_{11} & 0 & 0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44}
\end{array}
\right).
\label{eqn1}
\end{equation}

Non-zero entries occur only along the diagonal and anti-diagonal. Many calculations of entanglement and other properties \cite{ref4,ref5}, and their evolution under unitary or dissipative processes \cite{ref6}, can be carried out analytically for such states which make them appealing objects for study. Many other states of interest, such as the maximally entangled Bell states \cite{ref1} and ``Werner" states \cite{ref7}, are a sub-class of $X$-states, lending further importance to their study.

Yet, no firmer definition has been given of what makes a pure or mixed system an $X$-state. This Letter provides such a definition in terms of their invariance properties, that a particular symmetry group or algebra underlies them. Such an identification of an underlying symmetry helps to explain the analytical results while at the same time providing a well defined procedure for their preparation. Recognizing the symmetry also makes computations involving such states, such as unitary operations on them or evaluating concurrence or other measures of entanglement, straightforward and easily tractable. And, finally, the symmetry also opens the way for constructing other density matrices which may not visually appear as X but are nevertheless similar, states of a different rendering of the same algebraic symmetry. Since they differ in entanglement and separability considerations, they may prove useful for study.

\section{The subalgebra of $X$-states}

Positivity and other standard requirements of any density matrix make the $X$-states shown in Eq.~(\ref{eqn1}) a seven-parameter family. The diagonal elements of the density matrix are real so that, along with the trace being fixed at 1, three real parameters describe those diagonal entries. Hermiticity to guarantee real eigenvalues reduces the off-diagonal entries to two complex (say $\rho_{14}$ and $\rho_{23}$, with $\rho_{41}$ and $\rho_{32}$ their respective complex conjugates) or four real parameters for the total of seven real parameters.

The full two qubit system has the symmetry of the SU(4) group and its algebra $su(4)$. Fifteen operators, most conveniently rendered as fifteen linearly independent 4 $\times$ 4 matrices or as Pauli spinors/matrices of the two spins, together with the unit matrix, provide a complete description of the general system. There are, however, several subalgebras of $su(4)$. A series of recent papers have provided a geometrical description of their states and operators \cite{ref8,ref9,ref10,ref11}. In particular, one subalgebra, $su(2) \times su(2) \times u(1)$, of seven operators or matrices occurs in many physical systems in quantum optics and quantum information \cite{ref8,ref9}. This Letter presents them as the invariance set of the $X$-states.

Inspection of the explicit 4 $\times$ 4 matrices in a standard basis for two spins, $(|\uparrow \uparrow
\rangle, |\uparrow \downarrow \rangle, |\downarrow \uparrow \rangle,
|\downarrow \downarrow \rangle)$, is instructive \cite{ref8,ref9,ref12,ref13} and points immediately to sets of seven of them with the same structure of eight zeroes in the same positions as in Eq.~(\ref{eqn1}). That is, these are operators that do not mix the 1-4 and 2-3 subspaces of the density matrix. Combined with the observation that such a set of seven matrices closes under multiplication, it is immediate that they will carry $X$-states into each other, that they preserve the X structure. For this purpose, both the Lie algebra aspect that the seven operators close under commutation and their Clifford algebraic structure that they close under multiplication are important. Indeed, explicit rendering of the fifteen operators in terms of two Pauli spinors called $\vec{\sigma}$ and $\vec{\tau}$, together with the familiar multiplication rule $\sigma_i \sigma_j = \delta_{ij} + i \epsilon_{ijk} \sigma_k, i, j, k = 1-3$, where $\epsilon_{ijk}$ is the completely antisymmetric symbol and repeated indices are summed, is very useful for operations with them.

There are many such sets of seven operators/matrices constituting the $su(2) \times su(2) \times u(1)$ subalgebra \cite{ref8,ref9,ref11}. In each of them, one operator, the $u(1)$ element, commutes with all six of the others which themselves can be further subdivided as shown in \cite{ref8} into two sets of ``pseudospins", two sets of three which obey commutation relations of angular momentum within each set while all three of one set commute with all three of the other. Any one of the fifteen operators can serve as the commuting element because, as shown in a table in \cite{ref9}, each row has six zeroes so that each identifies such a $su(2) \times su(2) \times u(1)$ set. There are, therefore, fifteen non-equivalent such subalgebras.

We will designate such a set by $\{X_i\}, i=1, 2, \ldots, 7$, with $X_1$ the commuting element. One such is $(X_1 =\sigma_z \tau_z, X_2=\sigma_y \tau_x, X_3=\tau_z, X_4=-\sigma_y\tau_y, X_5=\sigma_x\tau_y, X_6=\sigma_z, X_7=\sigma_x\tau_x)$. This is the same set that occurs in the CNOT quantum logic gate constructed out of two Josephson junctions and was extensively studied in that context \cite{ref8}. It was also pointed out that it occurs in nuclear magnetic resonance when each spin is in an external magnetic field in the $z$-direction while being coupled to each other through scalar coupling $\vec{\sigma} \cdot \vec{\tau}$ and ``cross-coherences" $\sigma_x\tau_y$ and $\sigma_y\tau_x$. But a different choice for the commuting element $X_1$ gives another such subalgebra, and we will return to this in section IV. Each $X_i$ is traceless, Hermitian, and unitary, and its square is unity so that the eigenvalues are ($\pm 1, \pm 1$).

With any such set, $\{X_i\}$, the density matrix that remains invariant under their operations can be rendered as a linear superposition of them,

\begin{equation}
\rho =(I + \Sigma_i g_i X_i)/4,
\label{eqn2}
\end{equation}
in analogy to that for a single spin, $(I+\Sigma_i g_i \sigma_i)/2$. The seven real coefficients $g_i$ in Eq.~(\ref{eqn2}) parametrize $X$-states and are equivalent to the seven parameters in the density matrix in Eq.~(\ref{eqn1}):

\begin{eqnarray}
g_1 & = & (\rho_{11} +\rho_{44}) -(\rho_{22} +\rho_{33}), \nonumber \\
g_2 & = & 2i(\rho_{14}-\rho_{41}+\rho_{32}-\rho_{23}), \nonumber \\
g_3 & = & (\rho_{11} -\rho_{44}) -(\rho_{22} -\rho_{33}), \nonumber \\
g_4 & = & 2(\rho_{14}+\rho_{41}-\rho_{32}-\rho_{23}), \nonumber \\
g_5 & = & 2i(\rho_{14}-\rho_{41}-\rho_{32}+\rho_{23}), \nonumber \\
g_6 & = & (\rho_{11} -\rho_{44}) +(\rho_{22} -\rho_{33}), \nonumber \\
g_7 & = & 2(\rho_{14}+\rho_{41}+\rho_{32}+\rho_{23}).
\label{eqn3}
\end{eqnarray}

The algebra of the seven $\{X_i\}$ is most conveniently captured by Fig. 1 as has recently been pointed out \cite{ref11}. This figure occurs in projective geometry as the ``Fano Plane" \cite{ref14} and also is used to represent the multiplication table for octonions \cite{ref15}. Arranging the seven operators at the vertices, mid-points of sides and in-center of an equilateral triangle, the seven lines shown (including the inscribed circle) each carry three points, providing the multiplication rule for those $\{X_i\}$. The notation of arrows is also borrowed from octonions except that unlike them which have all seven lines arrowed, the three internal verticals are not in Fig. 1. On those lines, all three operators mutually commute, so that the product of two gives the third regardless of order. On the four arrowed lines, the operators mutually anticommute so that the product of two gives $(\pm i)$ times the third, with plus (minus) signs along (against) the sense of the arrow. For this purpose, each line is regarded as a closed loop with a continuously circulating arrow. The central element commutes with all six of the others. For each of those, there is one ``conjugate" element with which it commutes and four with which it anticommutes. All of this can be read off by merely glancing at Fig. 1 and will provide simple rules for their manipulation in the next section.


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\section{ Concluding remarks}
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In multipartite quantum systems, the monogamy is probably one of
the most important relation which imposes severe restriction on the
 structure of entanglement distributed
among many parties.   In this context, the main interest of this
paper was the monogamy property of quantum discord in three
qubit systems where the information is encoded in even and
odd Glauber coherent states. In particular, we investigated the influence of
 photon excitations  on the shareability of quantum discord between
the three optical modes of a quantum of ${\rm GHZ}$-type. We derived the
quantum discord deficit by evaluating analytically  the pairwise
correlations in terms of the photon excitation number and the
optical strength of Glauber coherent states. The symmetric quasi-${\rm GHZ}$ coherent
states follow the monogamy property for any photon excitation order.
We have also shown that the photon excitation of
antisymmetric quasi-${\rm GHZ}$ coherent
states reduces the violation of the monogamy property especially in states
involving Glauber coherent states with small amplitudes.\\



\noindent Finally, the investigation of the influence of photon excitations on the monogamy of quantum correlations in the states of {\rm GHZ}-type using
geometric based quantifiers such as Hilbert-Schmidt norm  or trace distance would be
interesting. In the other hand,  another significant issue
which deserves to be examined concerns the evolution of quantum discord under  the effect of subtracting photons on
the pairwise correlations in multipartite coherent states.










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\bibitem{ref4} See, for instance, S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, Phys. Rev. Lett. {\bf 87}, 050401 (2001); G. L, Kamta and A. F. Starace, {\it ibid} {\bf 88}, 107901 (2002); J. S. Pratt, {\it ibid} {\bf 93}, 237205 (2004); S. J. Gu, G. S. Tian, and H. Q. Lin, Phys. Rev. A {\bf 71}, 052322 (2005); J. Wang, H. Batelaan, J. Podany, and A. F. Starace, J. Phys. B {\bf 39}, 4343 (2006).

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the same invariance algebra, although its density matrix has no zero entries:

\begin{equation}
\rho = \! \frac{1}{4}I \! + \! \frac{1}{4}\left(
\begin{array}{cccc}
-g_4+g_6 & g_3-ig_2 & -ig_5 & g_1-g_7 \\
g_3+ig_2 & g_4+g_6 & g_1+g_7 & ig_5 \\
ig_5 & g_1+g_7 & g_4-g_6 & g_3+ig_2 \\
g_1-g_7 & -ig_5 & g_3-ig_2 & -g_4-g_6
\end{array}
\right).
\label{eqn5}
\end{equation}
Its preparation, by adding to $\rho$ a transformation under $X_1 (=\sigma_x \tau_x)$, and its manipulation or calculation of concurrence and other properties, all proceed as in Section III. Indeed, as can be seen from the choice of the two sets of $\{X_i\}$, it differs from the ``standard" set by a cyclic permutation of the indices $(x, y, z)$ which means two $\pi/2$ rotations of axes, first clockwise with respect to $y$ and then counter-clockwise with respect to an intermediate $z$ axis. Alternatively, this can be represented as the application of $\pi$ pulses to the spins. Clearly, the physics is unchanged with mere change of bases.

Entanglement and other investigations of these other sets will be of interest. A natural extension for further investigation is to analyze similarly sets of invariant states of other subalgebras of $su(4)$ such as $su(3)$ and $so(5)$ \cite{ref10}. The recognition of invariant sets opens a new window into such studies, focusing on what is essential and what are simply changes in bases and representations.


I thank Mr. Mazhar Ali, and Drs. Gernot Alber and Joseph Renes for discussions and for their hospitality at the Technical University, Darmstadt, during this work. This work was supported by the Alexander von Humboldt Foundation.


  